[Paper Review] Symmetries, Symmetry Breaking, Gauge Symmetries

Franco Strocchi|arXiv (Cornell University)|Feb 23, 2015

Quantum Mechanics and Applications6 references10 citations

TL;DR

This paper reinterprets symmetries, symmetry breaking, and gauge symmetries in quantum field theory through operational principles, showing that physical states and observables—defined by measurement protocols and expectation values—determine physical content. It demonstrates that global and local gauge symmetries, while not directly observable, manifest through superselection rules, parastatistics, and topological invariants like θ vacua, resolving philosophical concerns about their physical relevance.

ABSTRACT

The concepts of symmetry, symmetry breaking and gauge symmetries are discussed, their operational meaning being displayed by the observables {\em and} the (physical) states. For infinitely extended systems the states fall into physically disjoint {\em phases} characterized by their behavior at infinity or boundary conditions, encoded in the ground state, which provide the cause of symmetry breaking without contradicting Curie Principle. Global gauge symmetries, not seen by the observables, are nevertheless displayed by detectable properties of the states (superselected quantum numbers and parastatistics). Local gauge symmetries are not seen also by the physical states; they appear only in non-positive representations of field algebras. Their role at the Lagrangian level is merely to ensure the validity on the physical states of local Gauss laws, obeyed by the currents which generate the corresponding global gauge symmetries; they are responsible for most distinctive physical properties of gauge quantum field theories. The topological invariants of a local gauge group define superselected quantum numbers, which account for the $θ$ vacua.

Motivation & Objective

• To clarify the physical and operational meaning of symmetries, especially spontaneous symmetry breaking (SSB), in quantum field theories.
• To resolve philosophical concerns about gauge symmetries being non-empirical by showing their indirect but detectable physical consequences.
• To explain how global and local gauge symmetries manifest through superselected quantum numbers and topological invariants in infinite systems.
• To clarify the role of boundary conditions and phases in infinite systems as the cause of SSB, consistent with Curie’s principle.
• To demonstrate that determinism is preserved in gauge theories since only observables and states—both with deterministic evolution—are physically relevant.

Proposed method

• Operational definition of observables via experimental measurement apparatuses and their expectation values in physical states.
• Classification of states into physically disjoint phases due to incompatible preparation protocols, especially in infinitely extended systems.
• Use of the algebraic approach to quantum field theory, focusing on the C*-algebra of observables and its representations.
• Identification of superselected quantum numbers via the joint spectrum of invariant polynomials of gauge generators in the center of the observable algebra.
• Analysis of local gauge symmetries through their action on field algebras, showing they act non-trivially only in non-positive (unphysical) representations.
• Derivation of local Gauss laws as physical consequences of local gauge symmetry, ensuring current conservation on physical states.

Experimental results

Research questions

• RQ1 How can spontaneous symmetry breaking be consistently explained in quantum field theories without violating the Principle of Sufficient Reason?
• RQ2 What is the physical meaning of gauge symmetries if they are not visible in observables or physical states?
• RQ3 How do global and local gauge symmetries lead to detectable physical effects despite being unobservable at the level of observables?
• RQ4 What is the role of boundary conditions and infinite systems in the emergence of symmetry breaking?
• RQ5 How do topological invariants of the local gauge group give rise to θ vacua and explain chiral symmetry breaking in QCD?

Key findings

• Spontaneous symmetry breaking in infinite systems arises from boundary conditions encoded in the ground state, not from a choice of vacuum, and is consistent with Curie’s principle.
• Global gauge symmetries are not visible in observables but manifest through superselected quantum numbers and parastatistics, which can be operationally detected.
• Local gauge symmetries do not act non-trivially on physical states or observables but ensure the validity of local Gauss laws on physical states.
• Topological invariants of the local gauge group define elements in the center of local observable algebras, whose spectrum corresponds to the θ angles of θ vacua.
• The absence of Goldstone bosons in QCD is explained by the topological structure of the gauge group and the existence of θ vacua, not by Higgs mechanism.
• Determinism is preserved in gauge theories because only observables and states—both with deterministic evolution—are physically relevant, resolving concerns about non-determinism.